11 problems

Let $p$ and $\alpha$ be the parameters from the preceding theorem, let $\{\lambda_n\}$ be a sequence in $\big(0,\frac{1}{p^\alpha}\big)^\infty$ satisfying…

Let $L>0$. For $n=1,2,3,\ldots$, let $un$ be the positive roots determined by $tan(nun L)=tanh(nun L)$, and let $Phin(y)$ denote the corresponding eigenfunctions. Let $Thetam(y)$,…

Let $H$ be the separable Hilbert space under consideration. A subset of $H$ is $ell^1$-bounded if its coefficient sums are uniformly bounded with respect to some Riesz basis of $H$…

Let ${\overline{\mathcal R}}_1^N:=\{1\}\cup\{\sqrt{3}\,\mathcal C_k,\sqrt{3}\,\mathcal S_k:k\le N\}$ be the truncated $L_2$-normalized system, and let $G_N$ be its Gram matrix. The…

Let $[a_\ell,b_\ell)$, $\ell=1,\ldots,L$, be disjoint intervals contained in $[0,1)$, with … For a set of frequencies $\Lambda\subset\mathbb{Z}$ and a measurable set $S$, write…

Fix integers $m>1$ and $d>1$. For a nonempty set $S\subseteq\mathbb{Z}_{m}^{d}$, let $S_k$ and $S_{k,R_k}^{\ast}$ be the discrete and truncated fractal constructions defined in the…

Let $p\geq 3$ be prime, let $G_{p}=\mathbb{Z}_{p}^{2}$, and let $H_{p}=\mathbb{Z}_{p}\times\\{0\\}$. Define … Let $\rho(E_{p})$ denote the Riesz ratio of $E_p$. Divergence conjectu…

Let $G$ be a finite abelian group, $H\subseteq G$ a subgroup, and $E\subseteq G$ a subset. Suppose $E$ multi-tiles $G$ by $H$ at level at most $\ell$, has at most $k$ distinct tran…

Let $D=\\{x\in\mathbb{R}^{2}:\lVert x\rVert\leq 1\\}$ be the unit disc. A measurable set is a Riesz set if its $L^{2}$ space admits a Riesz basis of exponentials. Olevskii–Ulanovsk…

Shkalikov's Riesz basis conjecture. The periodic or antiperiodic system of normalized root functions associated with this potential should be a Riesz basis.

Zhang–Walter conjecture. The families of functions … are Riesz bases.